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Prior to the work of GM, Redi (1982) (see also Solomon 1971) noted that a symmetric component to the mixing tensor should be present in order to represent irreversible downgradient diffusive effects of various

subgrid-scale processes. The orientation of the diffusive flux is down the tracer gradient as it occurs along the

neutral directions. The result is to align the tracer parallel to the neutral direction in the process of dissipating

all tracer moments except the mean. Such diffusion will not affect locally referenced potential density. Therefore, isoneutral diffusion will not change the system’s available potential energy. More discussion of isoneutral diffusion, and references, can be found in the companion paper by Griffies et al. (1998, hereafter referred to as GGPLDS).

Gent and McWilliams stirring and Redi diffusion form a framework in which many coarse-resolution

ocean models parameterize the mixing of tracers. Currently, there is a great deal of energy focused on understanding the implications and relevance of this framework for simulating ocean circulation. There have been notable improvements in the simulations (e.g., Danabasoglu and McWilliams 1995; Hirst and McDougall 1996) and yet there have also been some rather tentative results (e.g., England 1995; England and Holloway 1996; Duffy et al. 1995). In addition to realistic coarse-model simulations with the GM and Redi parameterizations, there is an increasing number of theoretical and idealized studies aimed at clarifying certain of the conceptual issues (e.g., Held and Larichev 1996; Mc-Dougall and McIntosh 1996; Tandon and Garrett 1996; Holloway 1997; Treguier et al. 1997; Visbeck et al. 1997; Greatbatch 1998; Killworth 1998, Gille and Davis 1997, manuscript submitted to J. Phys. Oceanogr.; Dukowicz and Smith 1997).

This paper does not resolve any of the outstanding issues. Rather, it simply endeavors to bring the Redi and GM ideas onto an equal footing so that certain of their mathematical and physical properties can be directly compared and contrasted. The purpose of such an effort is twofold: First, the results presented here are arguably the simplest conceptual framework for thinking about the individual and combined effects of GM stirring and Redi diffusion (see also Holloway 1997). This framework may be useful when examining the effects of these subgrid-scale parameterizations on ocean density and tracer fields. Second, and most pragmatically, these results provide an almost trivial manner for which to implement GM and Redi in z-coordinate ocean models. The key element in this effort is the skew-diffusive flux (e.g., Plumb 1979; Moffatt 1983; Middleton and Loder 1989) arising from the GM closure. The perspective engendered by the GM skew flux provides some useful insights, which can be considered complementary to the more familiar advective flux formulation of GWMM. The plan of this paper is the following. General kinematical notions of skew fluxes are presented in section 2. Properties of the GM skew flux are discussed in section 3. The combined effects of GM skew diffusion and Redi diffusion are given in section 4, and numerical considerations are presented in section 5. Summary and conclusions are provided in section 6.

subgrid-scale processes. The orientation of the diffusive flux is down the tracer gradient as it occurs along the

neutral directions. The result is to align the tracer parallel to the neutral direction in the process of dissipating

all tracer moments except the mean. Such diffusion will not affect locally referenced potential density. Therefore, isoneutral diffusion will not change the system’s available potential energy. More discussion of isoneutral diffusion, and references, can be found in the companion paper by Griffies et al. (1998, hereafter referred to as GGPLDS).

Gent and McWilliams stirring and Redi diffusion form a framework in which many coarse-resolution

ocean models parameterize the mixing of tracers. Currently, there is a great deal of energy focused on understanding the implications and relevance of this framework for simulating ocean circulation. There have been notable improvements in the simulations (e.g., Danabasoglu and McWilliams 1995; Hirst and McDougall 1996) and yet there have also been some rather tentative results (e.g., England 1995; England and Holloway 1996; Duffy et al. 1995). In addition to realistic coarse-model simulations with the GM and Redi parameterizations, there is an increasing number of theoretical and idealized studies aimed at clarifying certain of the conceptual issues (e.g., Held and Larichev 1996; Mc-Dougall and McIntosh 1996; Tandon and Garrett 1996; Holloway 1997; Treguier et al. 1997; Visbeck et al. 1997; Greatbatch 1998; Killworth 1998, Gille and Davis 1997, manuscript submitted to J. Phys. Oceanogr.; Dukowicz and Smith 1997).

This paper does not resolve any of the outstanding issues. Rather, it simply endeavors to bring the Redi and GM ideas onto an equal footing so that certain of their mathematical and physical properties can be directly compared and contrasted. The purpose of such an effort is twofold: First, the results presented here are arguably the simplest conceptual framework for thinking about the individual and combined effects of GM stirring and Redi diffusion (see also Holloway 1997). This framework may be useful when examining the effects of these subgrid-scale parameterizations on ocean density and tracer fields. Second, and most pragmatically, these results provide an almost trivial manner for which to implement GM and Redi in z-coordinate ocean models. The key element in this effort is the skew-diffusive flux (e.g., Plumb 1979; Moffatt 1983; Middleton and Loder 1989) arising from the GM closure. The perspective engendered by the GM skew flux provides some useful insights, which can be considered complementary to the more familiar advective flux formulation of GWMM. The plan of this paper is the following. General kinematical notions of skew fluxes are presented in section 2. Properties of the GM skew flux are discussed in section 3. The combined effects of GM skew diffusion and Redi diffusion are given in section 4, and numerical considerations are presented in section 5. Summary and conclusions are provided in section 6.

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